a comparison of buckling analysis of the nanoplate and nanocomposite plate with central square hole embedded in winkler foundation is presented in this article. in order to enhance mechanical properties of nanoplate with central cut out, carbon nano tubes (cnts) are applied as uniform distribution (ud) through thickness direction. in order to define the shape function of plate with square cut o...

In this investigation Rayleigh-Ritz variational method has been applied to determine the least natural frequency coefficient for the title problem. Classical plate theory assumptions have been used to calculate strain energy and kinetic energy. Coordinate functions are combination of polynomials which satisfy boundary conditions at the outer boundary and trigonometric terms. In the second part ...

For a given subspace, the q-Rayleigh-Ritz method projects the large quadratic eigenvalue problem (QEP) onto it and produces a small sized dense QEP. Similar to the Rayleigh-Ritz method for the linear eigenvalue problem, the q-Rayleigh-Ritz method defines the q-Ritz values and the q-Ritz vectors of the QEP with respect to the projection subspace. We analyze the convergence of the method when the...

The following estimate for the Rayleigh–Ritz method is proved: |λ̃−λ||(ũ,u)| ≤ ‖Aũ− λ̃ũ‖sin∠{u;Ũ}, ‖u‖= 1. Here A is a bounded self-adjoint operator in a real Hilbert/euclidian space, {λ,u} one of its eigenpairs, Ũ a trial subspace for the Rayleigh–Ritz method, and {λ̃, ũ} a Ritz pair. This inequality makes it possible to analyze the fine structure of the error of the Rayleigh–Ritz method, in part...

We extend the Rayleigh–Ritz method to the eigen-problem of periodic matrix pairs. Assuming that the deviations of the desired periodic eigenvectors from the corresponding periodic subspaces tend to zero, we show that there exist periodic Ritz values that converge to the desired periodic eigenvalues unconditionally, yet the periodic Ritz vectorsmay fail to converge. To overcome this potential pr...

We consider the two-sided Arnoldi method and propose a two-sided Krylov–Schurtype restarting method. We discuss the restart for standard Rayleigh–Ritz extraction as well as harmonic Rayleigh–Ritz extraction. Additionally, we provide error bounds for Ritz values and Ritz vectors in the context of oblique projections and present generalizations of, e.g., the Bauer–Fike theorem and Saad’s theorem....

Composite stiffened cylindrical shells are widely used as primary elements in aerospace structures. In the recent years, there has been a growing research interest in optimum design of composite stiffened cylindrical shell structures for stability under buckling load.  This paper focuses upon the development of an efficient optimization of ring-stringer stiffened cylindrical shell. The opti...

The Rayleigh-Ritz method finds the stationary values, called Ritz values, of the Rayleigh quotient on a given trial subspace as approximations to eigenvalues of a Hermitian operator A. If the trial subspace is A-invariant, the Ritz values are exactly some of the eigenvalues of A. Given two subspaces X and Y of the same finite dimension, such that X is A-invariant, the absolute changes in the Ri...

using  principle  of  minimum  total  potential  energy  approach  in conjunction  with  rayleigh-ritz  method,  the  electro-thermo-mechanical  axial  buckling  behavior  of  piezoelectric  polymeric cylindrical  shell  reinforced  with  double-walled  boron-nitride nanotube  (dwbnnt)  is  investigated. coupling  between  electrical and mechanical  fields  are  considered  according  to  a  re...

This paper concerns a harmonic projection method for computing an approximation to an eigenpair (λ, x) of a large matrix A. Given a target point τ and a subspace W that contains an approximation to x, the harmonic projection method returns an approximation (μ + τ, x̃) to (λ, x). Three convergence results are established as the deviation of x from W approaches zero. First, the harmonic Ritz value...